Finite-element modeling of nano-indentation of thin-film materials
Abstract
Measuring mechanical properties of materials on a very small scale is a difficult, but
increasingly important task. There are only a few existing technologies for conducting
quantitative measurements of mechanical properties of nanostructures, and nanoindentation is the leading candidate. In this project, we simulate the nano-indentation
tests of thin film materials using finite element software ABAQUS.
The materials
properties and test parameters will be taken from references on nano-indentation
experiments [1, 2]. Therefore, the model can be validated by comparing its predictions
with experiment results. In addition, we will change 1) the thickness of the thin film and
2) the material of the substrate (for the thin film) in the model, in order to study
substrate’s effects on nano-indentation tests.
1. Introduction
The rapid development of nanostructured materials, thin film and coating
technologies has led to increasing demands for quantitative evaluation of their
mechanical properties. One of the methods widely used for such measurements is
depth-sensing indentation at low load, often termed ”nanoindentation”. This method
has been employed by many researchers to investigate the elastic and plastic
properties of various thin films [].
A standard experiment set-up for nanindetation of thin films is schematically depicted
in Figure 1 (a). During the nanoindetation process, an indenter is pressed against the
thin film, which is supported by a substrate.
Force Sensor
Indenter
Film
Substrate
(a)
Applied Load on Indenter P
Displacement
Sensor
dP/dh
Unloading
Loading
Displacement of Indenter h
(b)
Figure 1. The schematic plot of experiment set-up (a) and displacement-load curve (b)
for nanoindentation of
The load applied on the indenter and the indentation depth is recorded by two sensors
above the indenter. The results are then plotted as a load-displacement curve, one
example of which is shown on Figure 1(b). The slope of the curve upon unloading is
very important, because based on this slope the reduced Young’s modulus of the thin
film can be calculated. As the thin film is usually supported by a substrate during
nanoindetation, the load-displacement curve may be affected by the mechanical
properties of the substrate.
In this project, we will simulate the nanoindentation process of copper thin films
deposited on various substrates. The modeling results will be compared with
analytical solutions so as to validate the model. In addition we will discuss the effects
of substrate materials on nanoindetation of thin films.
2. Model Description
2.1. Model Geometry
The geometry of model developed in this project is schematically plotted in Figure 2.
The model consists of three parts: an indenter, the thin film to be tested, and the
substrate. All of them have axiasymetric geometries, and their dimensions are
indicated in Figure 2.
100 nm
1000 nm
100 nm
900 nm
Figure 2. Schematic plot of the nanoindetation model.
2.2. Materials Properties
The indenter is made from diamond. The thin film is made from copper. In order to
study the effect of substrates, we use 5 different materials for the substrate in the
model: sapphire, silicon, glass, copper, or polymer. The copper in the model is
elastoplastic, and its constitutive behavior is shown on Figure 3.
3
 (GPa)
2.5
2
Elastic
Plastic
1.5
1
0.5
0
0
0.01
0.02 0.03
 (-)
0.04
0.05
Figure 3. Constitutive behavior of copper
The other materials in the model are assumed to be perfect elastic, and their properties are
listed in Table 1.
Elastic
E
v (-)
Materials (GPa)
Diamond 1147
0.3
Sapphire 440
0.3
Silicon
172
0.3
Glass
73
0.3
Polymer 30
0.3
Table 1. Elastic properties of diamond, sapphire, silicon, glass, and polymer.
2.3. Finite Element Mesh
The model has been discretized into 4-noded axisymetric elements, as shown in
Figure 4.
Indenter
Film
Substrate
Figure 4. Constitutive behavior of copper
It can be seen the mesh around the indentation area is comparatively fine with
element dimension around 2nm. On the other hand, we use coarse mesh for the region
far away from indentation, with element dimension around 20 nm.
3. Model Validation
In order to validate the model, we use assign the material of the substrate to be copper.
Therefore, the model is equilent to nanoindentation of bulk copper. For
nanoindentation of bulk materials, the reduced modulus of the material can be
calculated analytically with the following equation:
Er 
1
1 v

 Es
2
s
  1  vi2 
  

  Ei 
in which Es and v s are the Young’s modulus and poisson ratio of the sample to be
tested, and E i and vi are those for the indenter.
On the other hand, the reduced modulus can also be measured experimentally with
the following formula:
 
Er   
4
1/ 2
 1

 Ac
1/ 2
  dP 
  
  dh  unloading
in which Ac is the contacting area between the indenter and the film, and
 dP 
is the slope of the load-displacement curve upon unloading.
 
 dh unloading
The simulated load-displacement curve and the analytical results are compared in
Figure 5. It can be seen the two curves match each other very well. Moreover, the
analytical reduced modulus is 128.2340(G Pa) , and the simulated one is
128.6056(G Pa) . Therefore, it proves our model can accurate simulate the
nanoindentation process.
Applied Load on Indenter P (nN)
4
14
x 10
Simulated P vs. h
Theoretical P vs. h
12
10
8
6
4
2
0
0
5
10
15
20
25
Displacement of Indenter h (nm)
30
Figure 5. Simulated load-displacement curve and the analytical results
4. Effect of Substrates
Now, we use different materials for the substrate and repeat the simulation. The loaddisplacement curve obtained are plotted in Figure 6 (a) and compared with the
experimental results in Figure 6 (b). It can be seen the simulated and experimental curves
have the same tendency: in order to reach the same indentation depth, a stiffer substrate
needs higher load, and a more compliant substrate needs lower load.
Force on Indenter P (nN)
4
x 10
18
E reducing
copper
16
sapphire
14
silicon
12
glass
10
polymer
8
6
4
2
0
0
5 10 15 20 25 30 35
Displacement of Indenter h (nm)
(a)
(b)
Figure 6. Simulated (a) and experimental (b) load-displacement curves.
In addition, the reduced modulus given by various substrates are shown in Figure 6 and
compared with experiment results. It’s obvious that a stiffer substrate gives higher
Reduced Youngs Modulus Er (GPa)
reduced modulus, and a more compliant gives a lower reduced modulus.
220
200
180
copper
sapp
silicon
glass
polymer
Theoretical Er
160
140
120
100
y=-0.00090793*x2
+0.75727*x+46.0508
80
60
0
50 100 150 200 250 300 350 400 450 500
Substrate Youngs Modulus E s(GPa)
(b)
(a)
Figure 6. Simulated (a) and experimental (b) load-displacement curves.
5. Conclusion
In conclusion, nanoindentation process can be simulated using finite element method.
The reduced modulus predicted by the finite-element model is very close to analytical
results. Stiff substrate tends to overestimate thin film’s modulus, and compliant substrate
tends to underestimate thin film’s modulus.
Reference
[1] X. Huang et al, Journal of Composite Materials, v40, p1393 (2006)
[2] J. Knapp et al, Journal of Applied Physics, v85, p1460 (1999)